The paper studies the case of shrinking cylindrical gas bubbles acting as a radial piston and generating acceleration waves. The behavior of such waves and their improbable transformation into shocks are illustrated theoretically, as well as through some examples inspired by experimental data. The use of rational extended thermodynamics enables us to highlight the relevance of the dissipation and the possible role played by dynamic pressure and stress tensor in bubble evolution or shock formation. These results constitute an extension and a completion of a previous work dedicated to the analysis of acceleration waves generated in oscillating spherical bubbles.